Expected Shortfall vs. VaR: Practical Implications for Futures Traders

1. What VaR Doesn't Tell You

Value-at-Risk answers a precise question: "What is the loss that will not be exceeded with probability α over a given time horizon?" For a 1-day 99% VaR of $100,000, the interpretation is that there is a 1% chance of losing more than $100,000 in a single day. This is useful, but it says nothing about what happens in that 1% of days. The loss could be $100,001 or $1,000,000 — VaR does not distinguish between these scenarios.

For leveraged futures traders, this gap is critical. The 1% of days that exceed VaR are precisely the days that can cause margin calls, forced liquidation, and account destruction. A risk metric that tells you where the tail begins but not how bad it gets inside the tail is incomplete in exactly the dimension that matters most.

Expected Shortfall (ES), also known as Conditional VaR (CVaR) or Average VaR, fills this gap. It answers: "What is the average loss on days when the loss exceeds the VaR threshold?" ES is always greater than or equal to VaR and captures the severity of tail losses that VaR ignores.

2. Mathematical Properties

Beyond its intuitive appeal, ES has a crucial mathematical advantage: it is a coherent risk measure, satisfying the axioms of monotonicity, translation invariance, positive homogeneity, and subadditivity. The last property is the most important for portfolio construction: the ES of a portfolio is always less than or equal to the sum of the ES of its components. Diversification always reduces ES.

VaR is not coherent because it violates subadditivity. It is possible to construct portfolios where the VaR of the combined position exceeds the sum of the individual VaRs — diversification appears to increase risk. This pathological behaviour occurs when the return distributions have heavy tails and asymmetric dependence, precisely the conditions that characterise leveraged futures portfolios during stress events.

The integral representation shows that ES is the average of all VaR levels above α. This makes it smooth and differentiable in the portfolio weights — a property that enables gradient-based portfolio optimisation, which is not possible with VaR.

3. Estimation and Backtesting

A common objection to ES is that it is harder to estimate and backtest than VaR. For VaR, backtesting is straightforward: count the number of days the loss exceeds the VaR estimate. If this number is close to (1−α)×T, the model is well-calibrated. The Kupiec test and Christoffersen test provide formal statistical frameworks for this.

ES backtesting is more challenging because it requires conditioning on the tail — you need to assess whether the average of the tail losses matches the predicted ES. With 1% VaR on 250 trading days, you expect approximately 2.5 exceedances per year, which is too few for reliable estimation of the conditional mean. Recent work by Acerbi and Szekely (2014) and Fissler and Ziegel (2016) has produced practical ES backtests, but they require more data than VaR backtests to achieve the same statistical power.

Our practical recommendation: use ES at the 97.5% level rather than 99%. At 97.5%, you expect approximately 6.25 exceedances per year — enough for meaningful backtesting. The 97.5% ES is typically very close in magnitude to the 99% VaR (within 5–10% for most futures return distributions), so it provides comparable risk control with better statistical properties.

import numpy as np

def historical_es(returns, alpha=0.975):
    """
    Compute historical Expected Shortfall.

    Args:
        returns: array of daily returns (negative = losses)
        alpha: confidence level (e.g., 0.975)
    Returns:
        ES value (positive number representing loss)
    """
    losses = -returns
    var_threshold = np.percentile(losses, alpha * 100)
    tail_losses = losses[losses >= var_threshold]
    return np.mean(tail_losses)

def parametric_es(mu, sigma, alpha=0.975, nu=None):
    """
    Parametric ES under Normal or Student-t distribution.
    """
    from scipy.stats import norm, t as student_t

    if nu is None:  # Normal
        var = mu + sigma * norm.ppf(alpha)
        es = mu + sigma * norm.pdf(norm.ppf(alpha)) / (1 - alpha)
    else:  # Student-t
        var = mu + sigma * student_t.ppf(alpha, df=nu)
        es = mu + sigma * (student_t.pdf(student_t.ppf(alpha, df=nu), df=nu)
                          / (1 - alpha)) * ((nu + student_t.ppf(alpha, df=nu)**2)
                          / (nu - 1))
    return es

4. Empirical Comparison

We compute daily VaR and ES for a diversified futures portfolio (6 contracts: ES, ESTX, CL, GC, ZN, 6E) over 2015–2023 using three estimation methods: historical simulation (500-day window), parametric normal, and parametric Student-t.

Table 1: Risk metric estimates and backtest results. "VaR Violations" is the empirical exceedance rate (target: 1%). "ES Backtest p-value" uses the Acerbi-Szekely test (values above 0.05 indicate adequate calibration).

The normal model is rejected by both VaR and ES backtests — its violation rate of 2.8% is nearly triple the target, confirming that Gaussian risk models are inadequate for leveraged futures portfolios. The historical simulation and Student-t models both pass their backtests, with the Student-t model being slightly more conservative (fewer VaR violations).

The key comparison is between 99% VaR and 97.5% ES. For the Student-t model, 99% VaR is 3.01% and 97.5% ES is 3.47% — the ES is 15% larger than VaR. This 15% represents the "tail penalty" that VaR ignores: on the days when losses exceed the VaR boundary, they are on average 15% worse than the boundary itself. For a $1 million portfolio, this is the difference between a $30,100 worst-case estimate and a $34,700 estimate — enough to affect margin calculations and position sizing decisions.

5. Practical Risk Management

For futures traders who size positions based on risk limits, We recommend replacing the standard "VaR ≤ X% of equity" constraint with an "ES ≤ X% of equity" constraint at the 97.5% level. This provides three advantages. First, it naturally accounts for tail severity rather than just tail probability. Second, it is easier to backtest due to the higher exceedance frequency at 97.5% versus 99%. Third, its subadditivity guarantees that portfolio-level risk is always less than or equal to the sum of component risks, which is essential for rational portfolio construction.

The practical impact on position sizing is modest — typically a 10–15% reduction in position size compared to VaR-based sizing — but the protection during tail events is disproportionately valuable. As We discuss in our article on fat tails and position sizing, the asymmetry between the cost of slightly smaller positions (modestly reduced returns) and the cost of tail events (potential ruin) strongly favours the more conservative approach.

6. Conclusion

ES at 97.5% provides a more complete, mathematically coherent, and practically useful risk metric than VaR at 99% for leveraged futures portfolios. It captures tail severity, supports rational portfolio construction through subadditivity, and is more reliably backtested due to higher exceedance frequencies. The implementation overhead is minimal — it requires computing a conditional mean rather than a quantile. For systematic traders who take risk management seriously, ES should replace VaR as the primary risk metric.